The Cost of the Characteristic Multivector: Exact 3 D Baseline and nD Complexity

Previous work using GAALOP to generate symbolically optimised 3D Characteristic Multivector (CM) rotor code showed that CM-based registration can achieve runtimes comparable to standard SVD-based methods for Absolute Orientation and ICP, while retaining the accuracy behaviour reported for CM in earlier studies. In that work it was also observed that naive CM implementations based on generic multivector libraries tend to run slower than both SVD-based solvers and specialised CM code. In this paper we give an implementation level analysis of the computational cost of the CM rotor providing an intrinsic cost baseline that helps explain these empirical differences and that can guide the design and evaluation of efficient CM implementations. Although the CM construction applies in any dimension and signature with a non-degenerate metric and can be used with non-orthonormal frames, we specialise to the Euclidean orthonormal case Cℓ(n, 0) where n is the space dimension to make storage, constants and operation counts explicit. Our central idea is a stage-by-stage model in which each term is assigned to the stage determined by the largest frame index it contains. At stage i only coefficients indexed by even subsets of {1, ⋯, i} can change, so the main counts depend only on the dimension and not on the data. We derive closed form expressions for arithmetic and memory costs and compare two update strategies: an in-place scheme that writes each contribution directly to the stored even coefficients and a split scheme that collects all contributions for a stage in a temporary table and then commits them together. The main counts grow in direct proportion to the size N of the even subspace (N=2n-1 in Cℓ(n, 0)) with normalization contributing the same order but with a smaller constant. A complete 3D worked example fixes the constants and validates the model, and the resulting formulas provide a cost baseline for specialised implementations of the CM rotor that do not rely on generic multivector operations.